Difference between a mist pass and a normalized z-buffer?
Some differences:

The Mist pass is antialiased. The Z pass is not.

The range for the mist pass is set independently of the camera's clip point.


(To visualize the mist pass select the camera and enable mist on the Viewport display settings)

Start sets the black (0) portions of the image. Depth sets the range in which the mist pass will convert the distance to a grayscale. Past that range everyting is white (1).


The falloff on the mist pass can be changed to quadratic, linear or Inverse Quadratic. The Z pass is always linear.



Comparison
If you set the clip distance and a mist pass to the same range and set the falloff to linear, the only difference is the antialiasing.